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Properties

Label 225.8.b.g
Level $225$
Weight $8$
Character orbit 225.b
Analytic conductor $70.287$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [225,8,Mod(199,225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 8, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("225.199"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 225.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,206,0,0,0,0,0,0,-16900] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(70.2866307339\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 5i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 103 q^{4} - 186 \beta q^{7} + 231 \beta q^{8} - 8450 q^{11} + 1244 \beta q^{13} + 4650 q^{14} + 7409 q^{16} + 1918 \beta q^{17} + 45884 q^{19} - 8450 \beta q^{22} - 20424 \beta q^{23} + \cdots - 41357 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 206 q^{4} - 16900 q^{11} + 9300 q^{14} + 14818 q^{16} + 91768 q^{19} - 62200 q^{26} + 175100 q^{29} - 152424 q^{31} - 95900 q^{34} - 207200 q^{41} - 1740700 q^{44} + 1021200 q^{46} - 82714 q^{49}+ \cdots + 8558800 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
− 1.00000i
1.00000i
− 5.00000i 0 103.000 0 0 930.000i − 1155.00i 0 0
199.2 5.00000i 0 103.000 0 0 − 930.000i 1155.00i 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.8.b.g 2
3.b odd 2 1 225.8.b.h 2
5.b even 2 1 inner 225.8.b.g 2
5.c odd 4 1 45.8.a.b ✓ 1
5.c odd 4 1 225.8.a.h 1
15.d odd 2 1 225.8.b.h 2
15.e even 4 1 45.8.a.c yes 1
15.e even 4 1 225.8.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.8.a.b ✓ 1 5.c odd 4 1
45.8.a.c yes 1 15.e even 4 1
225.8.a.e 1 15.e even 4 1
225.8.a.h 1 5.c odd 4 1
225.8.b.g 2 1.a even 1 1 trivial
225.8.b.g 2 5.b even 2 1 inner
225.8.b.h 2 3.b odd 2 1
225.8.b.h 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(225, [\chi])\):

\( T_{2}^{2} + 25 \) Copy content Toggle raw display
\( T_{11} + 8450 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 25 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 864900 \) Copy content Toggle raw display
$11$ \( (T + 8450)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 38688400 \) Copy content Toggle raw display
$17$ \( T^{2} + 91968100 \) Copy content Toggle raw display
$19$ \( (T - 45884)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 10428494400 \) Copy content Toggle raw display
$29$ \( (T - 87550)^{2} \) Copy content Toggle raw display
$31$ \( (T + 76212)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 69928513600 \) Copy content Toggle raw display
$41$ \( (T + 103600)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 105417102400 \) Copy content Toggle raw display
$47$ \( T^{2} + 732530574400 \) Copy content Toggle raw display
$53$ \( T^{2} + 918128076100 \) Copy content Toggle raw display
$59$ \( (T + 1239550)^{2} \) Copy content Toggle raw display
$61$ \( (T - 628522)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 96335744400 \) Copy content Toggle raw display
$71$ \( (T + 3934300)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 20757956088100 \) Copy content Toggle raw display
$79$ \( (T + 5371644)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 45038326323600 \) Copy content Toggle raw display
$89$ \( (T - 3346500)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 250580351872900 \) Copy content Toggle raw display
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